Transient solutions for three‐dimensional lid‐driven cavity flows by a least‐squares finite element method

A time‐accurate least‐squares finite element method is used to simulate three‐dimensional flows in a cubic cavity with a uniform moving top. The time‐ accurate solutions are obtained by the Crank‐Nicolson method for time integration and Newton linearization for the convective terms with extensive linearization steps. A matrix‐free algorithm of the Jacobi conjugate gradient method is used to solve the symmetric, positive definite linear system of equations. To show that the least‐squares finite element method with the Jacobi conjugate gradient technique has promising potential to provide implicit, fully coupled and time‐accurate solutions to large‐scale three‐dimensional fluid flows, we present results for three‐dimensional lid‐driven flows in a cubic cavity for Reynolds numbers up to 3200.